Constrained optimization. As mentioned, there is a growing body of literature on designing various StoSQP schemes for solving Problem (REF ). Compared to penalty methods and augmented Lagrangian methods, SQP methods preserve the problem structure, are more robust to initialization, and do not suffer ill-conditioning issues.
[1]} designed a very first StoSQP scheme. At each iteration, the scheme adaptively selects a penalty parameter of the \(\ell _1\)  merit function to ensure the Newton direction generates a sufficient decrease on the merit function; and then selects a random stepsize \({\bar{\alpha }}_t\)  based on sequences \(\beta _t\)  and \(\chi _t = O(\beta _t^2)\) , such that \(\beta _t\le {\bar{\alpha }}_t\le \eta _t=\beta _t+\chi _t\) .
An alternative StoSQP scheme is designed in [2]}, where the authors embedded stochastic line search into StoSQP to get rid of the prespecified sequences \(\beta _t, \chi _t\) . That algorithm is more adaptive than the algorithm of [1]}, while requiring a more stringent setup for line search—one has to generate batch samples with an increasing batch size in each iteration.
Subsequently, [4]} designed a StoSQP to remove constraint qualification condition; [5]} designed an inexact StoSQP to solve Newton systems approximately; [6]} designed an active-set StoSQP to enable inequality constraints; [7]} designed an accelerated StoSQP by applying variance reduction technique on finite-sum objective.
All these works showed global convergence of different StoSQP schemes. In addition to these works, [8]}, [9]} considered optimization with noisy functions. Those analyses require known, deterministic, and bounded noise, and thus they are not suited for the considered problems in (REF ).
